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Probl. Peredachi Inf., 2008, Volume 44, Issue 2, Pages 75–95 (Mi ppi1272)  

This article is cited in 7 scientific papers (total in 7 papers)

Large Systems

Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove results on exact asymptotics of the probabilities
$$ \mathrm{P}\{\int_0^1|\eta(t)|^p dt\leq\varepsilon^p\},\quad\varepsilon\to 0, $$
where $2\leq p\leq\infty$, for two types of Gaussian processes $\eta(t)$, namely, a stationary Ornstein–Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation
\begin{gather*} dZ(t)=dw(t)+g(t)Z(t)dt,\quad t\in[0,1],
Z(0)=0. \end{gather*}
Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov's absolute continuity theorem.

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English version:
Problems of Information Transmission, 2008, 44:2, 138–155

Bibliographic databases:

UDC: 621.391.1:519.2
Received: 29.11.2007

Citation: V. R. Fatalov, “Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$”, Probl. Peredachi Inf., 44:2 (2008), 75–95; Problems Inform. Transmission, 44:2 (2008), 138–155

Citation in format AMSBIB
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\pages 75--95
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. R. Fatalov, “Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$”, Problems Inform. Transmission, 46:1 (2010), 62–85  mathnet  crossref  mathscinet  isi
    2. Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81  mathnet  crossref  crossref  zmath  isi  elib  elib
    3. V. R. Fatalov, “Asymptotic behavior of small deviations for Bogoliubov's Gaussian measure in the $L^p$ norm, $2\le p\le\infty$”, Theoret. and Math. Phys., 173:3 (2012), 1720–1733  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    4. V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. V. R. Fatalov, “Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$”, Problems Inform. Transmission, 50:4 (2014), 371–389  mathnet  crossref  isi
    6. V. R. Fatalov, “Weighted $L^p$, $p\ge2$, for a wiener process: Exact asymptoties of small deviations”, Moscow University Mathematics Bulletin, 70:2 (2015), 68–73  mathnet  crossref  mathscinet  isi
    7. Lototsky S.V., “Small Ball Probabilities For the Infinite-Dimensional Ornstein-Uhlenbeck Process in Sobolev Spaces”, Stoch. Partial Differ. Equ.-Anal. Comput., 5:2 (2017), 192–219  crossref  mathscinet  zmath  isi
  • Проблемы передачи информации Problems of Information Transmission
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